SUBSTITUTION, a very common algebraical process, being, as its name imports, the substituting for any quantity another which is equal to it.
A method of approximation, which is frequently used and of great importance, has obtained the name of successive substitution. Let any equation be rednced to the form x = a ÷ eetr, where e is less than unity, and cpx a function of x. If we make x= a, the error thereby committed is less than cpx, being cox, iu which e is less than unity. Take this value x = a, and substitute it on the second side, giving x = a + c¢a : this value is nearer than the last in most cases, for it should be x = a - eo (a - - cox) =a + epa ectia. cpx nearly, where Va is the differential coefficient, of oa. The last error was erpx, and the present error is less, if eitla be less than unity. Gene rally, if fur x we write a +p, and if p be erroneous by a quantity of the order e , we shall hue, by one more substitution, x a + CO(a(a+ p).
Now the error of Ca 4-p) will be of the order e and that of op (e +p) of the order e'+'. There is then a continual approximation to the value of .r.
with x= a + eget, in which the error is of the order et, we have x = a + (0 + in which the error is of the third order. Rejecting terms of the third order, we have x = + e•pa + eltpals'a.
Substitute this again, and we have x= a + eip a + eta + esPastla in which the error Is of the fourth order. Rejecting terms of the fourth order, x es a + spa + eletass'a + j (Carl>. + 2— (#a)r and so on: the development being made by TAYLOR'S THEOREM. This would lead in effect to the celebrated theorem of Lagrange ; but the actual method of substitution is sometimes preferable.